55.1.13 problem HW 5 problem 5

Internal problem ID [8709]
Book : Selected problems from homeworks from different courses
Section : Math 2520, summer 2021. Differential Equations and Linear Algebra. Normandale college, Bloomington, Minnesota
Problem number : HW 5 problem 5
Date solved : Sunday, March 30, 2025 at 01:24:17 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+5 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = -2\\ y \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 31

ode:=[diff(x(t),t) = -2*x(t)+3*y(t), diff(y(t),t) = -2*x(t)+5*y(t)]; 
ic:=x(0) = -2y(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{-t}+{\mathrm e}^{4 t} \\ y \left (t \right ) &= -{\mathrm e}^{-t}+2 \,{\mathrm e}^{4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 36

ode={D[x[t],t]==-2*x[t]+3*y[t],D[y[t],t]==-2*x[t]+5*y[t]}; 
ic={x[0]==-2,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} \left (e^{5 t}-3\right ) \\ y(t)\to e^{-t} \left (2 e^{5 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 3 C_{1} e^{- t} + \frac {C_{2} e^{4 t}}{2}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{4 t}\right ] \]